有界噪声参激下Duffing振子的混沌运动

本文研究有界噪声参激下Ｄｕｆｆｉｇ振子出现混沌运动的可能性。首先推导了随机Ｍｅｌｎｉｋｏｖ过程,由广义过程在均方意义上出现简单零点给出了可能出现混沌的临界激励幅值,其次用数值方法计算了该系统的最大Ｌｙａｐｕｎｏｖ指数,由最大Ｌｙａｐｕｎｏｖ指数为零,给出了出现混沌的另一个临界激励幅值,发现在噪声强度大于一定值后,两个临界幅值均随噪声强度的增大而增大。     

van der Pol振子对高斯白噪声的响应

在过去20多年中,若干作者曾用不同方法研究过van der Pol振子对高斯白噪声的稳态响应,得到了不同甚至相反的结论。本文对此问题作了更为深入的研究,结果表明,在非线性不大时,随机平均法与等效非线性微分方程法给出良好的近似,而等效线性化法与高斯截断法给出错误的结论。     

联合噪声激励下分数阶三稳van der Pol振子的随机P分岔

研究了联合高斯白噪声激励下含分数阶导数项的三稳态van der Pol系统的随机P分岔问题.利用均方误差最小原则,将分数阶导数项等效为阻尼力与回复力的线性组合,从而将原系统转化为等价的整数阶系统.运用随机平均法得到了系统幅值的稳态概率密度函数(PDF),利用奇异性理论,得到了系统发生随机P分岔的临界参数条件.在转迁集曲...     

非高斯噪声中随机信号的渐近最优检测

本文研究了非高斯噪声中随机信号的检测问题。基于随机信号的参数模型和广义似然比检测理论,导出了非高斯噪声中随机信号Rao检测的数学解析式,其检测性能渐近等同于广义似然比检测但计算更有效。仿真结果表明,该检测器性能大大优于传统的能量检测器以及高斯噪声假设下的广义似然比检测器。     

色噪声激励下Duffing—Rayleigh—Mathieu系统的稳态响应

基于广义谐和函数与随机平均原理,研究了具有强非线性的Duffing-Rayleigh-Mathieu系统在色噪声激励下的稳态响应.通过van der Pol坐标变换,将系统运动方程转化为关于幅值与初始相位角的随机微分方程.应用Stratonovich-Khasminskii极限定理,作随机平均,得到近似的二维扩散过程.在此基础上,考虑共振情形,引入相位差变量,做确定性的平均,得到关于幅值与相位差的It(o)随机微分方程.建立对应的Fokker-Planck-Kolmogorov(FPK)方程,结合边界条件与归一化条件,用Crank-Nicolson型有限差分法求解稳态的FPK方程,得到平稳状态下系统的联合概率分布.用Monte Carlo数值模拟法验证了理论方法的有效性.     

非高斯噪声激励下非线性漂移Fokker-Planck方程的非稳态解及其应用

非高斯噪声广泛存在于各种非线性系统,对非高斯噪声所驱动系统的非稳态演化行为进行研究可以更为深入的了解其内在的演化机理．本文对非高斯噪声和高斯白噪声共同驱动的非线性动力学系统的非稳态演化问题进行研究．首先应用格林函数的 $Omega$ 展开理论在初始区域对非线性动力学系统进行线性化,然后结合本征值和本征矢理论推导出了该系统 Fokker-Planck 方程的近似非稳态解的表达式,最后以 Logistic 系统模型为例分析了非高斯噪声强度,关联时间及非高斯噪声偏离参数对非稳态解以及一阶矩的影响．研究结果表明,用 Logistic 模型描述产品产量增长时,其非稳态解可更好地反映产品产量在不稳定点附近的演化行为．     

混合高斯噪声中随机信号的最佳检测

根据加性窄带混合高斯噪声中随机参量信号最佳阈值检测模型构建了非高斯最佳检测器,并通过仿真试验验证了模型的有效性。     

高斯色噪声下的子阵平滑主模式抑制波束形成

实际海洋波导是一种具有空-时相关性的衰落信道,且其驻留的海洋环境背景噪声场呈现非均匀各向异性的特性,噪声场时空相关性并非δ函数,大孔径声呐的探测性能会受到很大影响。将背景噪声建模为有色高斯随机过程,设计基于子阵平滑的主模式抑制(Dominant Mode Rejection, DMR)波束形成算法,以实现小快拍数条件下具有去相关作用的自适应处理,较好地提高了大孔径阵声呐在高斯色噪声环境下的探测能力。模拟仿真结果表明,该方法具有良好的抑制高斯色噪声的能力。     

二阶随机系统的Lyapunov指数与稳定性

利用线性变换方法研究了二阶系统在随机扰动下系统的运动稳定性及分叉问题。给出了线性化系统最大Ｌｙａｐｕｎｏｖ指数的计算公式，从而由其最大Ｌｙａｐｕｎｏｖ指数为零可求出线性化系统几乎必然稳定区域的边界。     

非高斯噪声激励下分段非线性系统的平均首次穿越时间

噪声诱导的逃逸问题出现在众多研究领域,平均首次穿越时间作为用来表征粒子逃逸现象的重要特征量,现已被广泛应用于电子器件的开关时间及双稳器件的寿命等问题的研究之中.本文研究了由乘性非高斯噪声和加性高斯白噪声共同驱动下分段非线性系统的平均首次穿越时间问题.运用路径积分法、统一色噪声近似和最速下降法,得到了系统平均首次穿越时间的表达式.通过数值计算发现,在非高斯噪声偏离参数、噪声关联时间和互关联强度的作用下,非高斯噪声强度的增加会导致平均首次穿越时间曲线出现单峰结构,而加性噪声强度的增加会导致平均首次穿越时间的单调减小,这表明在该模型中非高斯噪声和高斯噪声对平均首次穿越时间的影响是不同的.此外还进一步讨论了非高斯噪声偏离参数、噪声关联时间和噪声互关联强度对平均首次穿越时间的影响.     

Van der Pol oscillator with time variable parameters

In this paper, the analytical solving procedure of the oscillator with slow time variable mass is developed. The solution is based on the Jacobi elliptic function whose properties: frequency, amplitude and modulus are obtained according to the requirements given for the amplitude of the displacement and the amplitude of the velocity of vibration and also period of vibration. The suggested procedure is applied for the solution of the time variable Van der Pol oscillator. The limit value of the initial mass of the oscillator is determined which separates the case when the limit cycle motion occurs, and the case when the amplitude of vibration tends to zero independently of the initial displacement. A numerical example is considered. The analytical solution is compared with the numerically obtained one and it is concluded that they are in good agreement.     

Asymptotic Lyapunov stability with probability one of Duffing oscillator subject to time-delayed feedback control and bounded noise excitation

The asymptotic Lyapunov stability with probability one of a Duffing system with time-delayed feedback control under bounded noise parametric excitation is studied. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method and the expression for the Lyapunov exponent of the linearized averaged Itô equations is derived. It is inferred that the Lyapunov exponent so obtained is the first approximation of the largest Lyapunov exponent of the original system, and the asymptotic Lyapunov stability with probability one of the original system can be determined approximately by using the Lyapunov exponent. Finally, the effects of time delay in feedback control on the Lyapunov exponent and the stability of the system are analyzed. The theoretical results are well verified through digital simulation.     

Response analysis of Van der Pol oscillator subjected to harmonic and random excitations

The response of the Van der Pol oscillator subjected to combined harmonic and random excitations is investigated by a technique combining two excellent methods, namely the stochastic averaging method and the equivalent linearization method. The original equation is averaged by the stochastic averaging method at first. Then the resulting nonlinear averaged equations are linearized by the equivalent linearization method so that the equations obtained can be solved exactly by the technique of auxiliary function. The mean square response of the Van der Pol oscillator is computed algebraically and compared to the ones obtained by numerical simulation and by technique combining methods of stochastic averaging and equivalent non-linearization. The results show that the proposed technique gives a good prediction on mean square responses of the Van der Pol oscillator. In addition, the proposed technique can be applied to other nonlinear systems for it does not require any special conditions, and it can be improved by advanced optimization criteria of the equivalent linearization method.     

Stochastic stability of Duffing oscillator with fractional derivative damping under combined harmonic and white noise parametric excitations

The stochastic stability of a Duffing oscillator with fractional derivative damping of order α (0 < α < 1) under parametric excitation of both harmonic and white noise is studied. First, the averaged Itô equations are derived by using the stochastic averaging method for an SDOF strongly nonlinear stochastic system with fractional derivative damping under combined harmonic and white noise excitations. Then, the expression for the largest Lyapunov exponent of the linearized averaged Itô equations is obtained and the asymptotic Lyapunov stability with probability one of the original system is determined approximately by using the largest Lyapunov exponent. Finally, the analytical results are confirmed by using those from a Monte Carlo simulation of the original system.     

The Duffing oscillator under combined periodic and random excitations

The Duffing oscillator under combined periodic and random excitations is investigated by a simple technique. The system response is separated into the deterministic and random parts governed by two coupled differential equations. The couple relation is expressed through varying on time coefficients which are approximately replaced by their averaging values over one period. This simplification yields that the two coupled differential equations can be solved by averaging and equivalent linearization methods. The mean-square response of the system is compared with the numerical results obtained by the finite element and Monte Carlo simulation methods. The results obtained show the interaction between the periodic and random excitations on the system response.     

Probabilistic response of an elastic perfectly plastic oscillator under Gaussian white noise

The response of an elastic perfectly plastic oscillator under zero mean Gaussian white noise excitation is studied in this paper. Considering the works of previous studies, a closed form expression of the mean maximum of the plastic drift is given assuming that the plastic process is equivalent to a Brownian motion. In order to better describe the plastic drift a probabilistic model is proposed for the yield increments which occur in clumps. To estimate the input parameters of this model, three methods, based on numerical computations of some relevant integrals, are presented. Alternatively, these parameters can be estimated, more conveniently, according to the results obtained more recently in the literature with the Slepian model approach. The results of numerical simulations show a quite satisfactory agreement with theoretical predictions.     

Noise-induced transition between stationary states of a van der Pol oscillator

It is shown that the white-noise-induced transition between the limiting cycle and state of rest of a van der Pol oscillator has a threshold. The threshold value is directly proportional to the product of the characteristic energy of self-oscillations and the friction coefficient. It is shown that not only self-oscillations, but also external noise, disappear in the state of rest.